Sylvester’s problem for beta-type distributions

Consider $d+2$ i.i.d. random points $X_1,\dots ,X_{d+2}$ in $R^d$. In this note, we compute the probability that their convex hull is a simplex focusing on three specific distributional settings:

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  [(i)] the distribution of $X_1$ is multivariate standard normal.
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  [(ii)] the density of $X_1$ is proportional to ${(1-{\Vert x\Vert }^2)}^{\beta }$ on the unit ball (the beta distribution).
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  [(iii)] the density of $X_1$ is proportional to ${(1+{\Vert x\Vert }^2)}^{-\beta }$ (the beta prime distribution).

In the Gaussian case, we show that this probability equals twice the sum of the solid angles of a regular $(d+1)$-dimensional simplex.